# 11.2 Graphing linear equations  (Page 4/6)

 Page 4 / 6

Graph the equation $y=-1.$

## Solution

The equation $y=-1$ has only variable, $y.$ The value of $y$ is constant. All the ordered pairs in the table have the same $y$ -coordinate, $-1$ . We choose $0,3,$ and $-3$ as values for $x.$

$y=-1$
$x$ $y$ $\left(x,y\right)$
$-3$ $-1$ $\left(-3,-1\right)$
$0$ $-1$ $\left(0,-1\right)$
$3$ $-1$ $\left(3,-1\right)$

The graph is a horizontal line passing through the $y$ -axis at $–1$ as shown.

Graph the equation: $y=-4.$

Graph the equation: $y=3.$

The equations for vertical and horizontal lines look very similar to equations like $y=4x.$ What is the difference between the equations $y=4x$ and $y=4?$

The equation $y=4x$ has both $x$ and $y.$ The value of $y$ depends on the value of $x.$ The $y\text{-coordinate}$ changes according to the value of $x.$

The equation $y=4$ has only one variable. The value of $y$ is constant. The $y\text{-coordinate}$ is always $4.$ It does not depend on the value of $x.$

The graph shows both equations.

Notice that the equation $y=4x$ gives a slanted line whereas $y=4$ gives a horizontal line.

Graph $y=-3x$ and $y=-3$ in the same rectangular coordinate system.

## Solution

Find three solutions for each equation. Notice that the first equation has the variable $x,$ while the second does not. Solutions for both equations are listed.

The graph shows both equations.

Graph the equations in the same rectangular coordinate system: $y=-4x$ and $y=-4.$

Graph the equations in the same rectangular coordinate system: $y=3$ and $y=3x.$

## Key concepts

• Graph a linear equation by plotting points.
1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
2. Plot the points on a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
3. Draw the line through the points. Extend the line to fill the grid and put arrows on both ends of the line.
• Graph of a Linear Equation: The graph of a linear equation $ax+by=c$ is a straight line.
• Every point on the line is a solution of the equation.
• Every solution of this equation is a point on this line.
• A linear equation can be graphed by finding ordered pairs that represent solutions, plotting them on a coordinate grid, and drawing a line through them. See [link] .
• A linear equation forms a line when the solutions are plotted on a coordinate grid. All of the solutions are on the line, and any points that are not on the line are not solutions.
• A vertical line is a line that goes up and down on a coordinate grid. The $x\text{-coordinates}$ of a vertical line are all the same. See [link] .
• A horizontal line is a line that goes sideways on a coordinate grid. The $y\text{-coordinates}$ of a vertical line are all the same. See [link] .

## Practice makes perfect

Recognize the Relation Between the Solutions of an Equation and its Graph

For each ordered pair, decide

1. is the ordered pair a solution to the equation?
2. is the point on the line?

$y=x+2$

1. $\left(0,2\right)$
2. $\left(1,2\right)$
3. $\left(-1,1\right)$
4. $\left(-3,1\right)$

1. yes yes
2. no no
3. yes yes
4. yes yes

$y=x-4$

1. $\left(0,-4\right)$
2. $\left(3,-1\right)$
3. $\left(2,2\right)$
4. $\left(1,-5\right)$

1. yes yes
2. yes yes
3. no no
4. no no

$y=\frac{1}{2}x-3$

1. $\left(0,-3\right)$
2. $\left(2,-2\right)$
3. $\left(-2,-4\right)$
4. $\left(4,1\right)$

1. yes yes
2. yes yes
3. yes yes
4. no no

$y=\frac{1}{3}x+2$

1. $\left(0,2\right)$
2. $\left(3,3\right)$
3. $\left(-3,2\right)$
4. $\left(-6,0\right)$

1. yes yes
2. yes yes
3. no no
4. yes yes

Graph a Linear Equation by Plotting Points

In the following exercises, graph by plotting points.

$y=3x-1$

$y=2x+3$

$y=-2x+2$

$y=-3x+1$

$y=x+2$

$y=x-3$

$y=-x-3$

$y=-x-2$

$y=2x$

$y=3x$

$y=-4x$

$y=-2x$

$y=\frac{1}{2}x+2$

$y=\frac{1}{3}x-1$

$y=\frac{4}{3}x-5$

$y=\frac{3}{2}x-3$

$y=-\frac{2}{5}x+1$

$y=-\frac{4}{5}x-1$

$y=-\frac{3}{2}x+2$

$y=-\frac{5}{3}x+4$

$x+y=6$

$x+y=4$

$x+y=-3$

$x+y=-2$

$x-y=2$

$x-y=1$

$x-y=-1$

$x-y=-3$

$-x+y=4$

$-x+y=3$

$-x-y=5$

$-x-y=1$

$3x+y=7$

$5x+y=6$

$2x+y=-3$

$4x+y=-5$

$2x+3y=12$

$3x-4y=12$

$\frac{1}{3}x+y=2$

$\frac{1}{2}x+y=3$

Graph Vertical and Horizontal lines

In the following exercises, graph the vertical and horizontal lines.

$x=4$

$x=3$

$x=-2$

$x=-5$

$y=3$

$y=1$

$y=-5$

$y=-2$

$x=\frac{7}{3}$

$x=\frac{5}{4}$

In the following exercises, graph each pair of equations in the same rectangular coordinate system.

$y=-\frac{1}{2}x$ and $y=-\frac{1}{2}$

$y=-\frac{1}{3}x$ and $y=-\frac{1}{3}$

$y=2x$ and $y=2$

$y=5x$ and $y=5$

Mixed Practice

In the following exercises, graph each equation.

$y=4x$

$y=2x$

$y=-\frac{1}{2}x+3$

$y=\frac{1}{4}x-2$

$y=-x$

$y=x$

$x-y=3$

$x+y=-5$

$4x+y=2$

$2x+y=6$

$y=-1$

$y=5$

$2x+6y=12$

$5x+2y=10$

$x=3$

$x=-4$

## Everyday math

Motor home cost The Robinsons rented a motor home for one week to go on vacation. It cost them $\text{594}$ plus $\text{0.32}$ per mile to rent the motor home, so the linear equation $y=594+0.32x$ gives the cost, $y,$ for driving $x$ miles. Calculate the rental cost for driving $400,800,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}1,200$ miles, and then graph the line.

$722,$850, \$978

Weekly earning At the art gallery where he works, Salvador gets paid $\text{200}$ per week plus $\text{15%}$ of the sales he makes, so the equation $y=200+0.15x$ gives the amount $y$ he earns for selling $x$ dollars of artwork. Calculate the amount Salvador earns for selling $\text{900, 1,600},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{2,000},$ and then graph the line.

## Writing exercises

Explain how you would choose three $x\text{-values}$ to make a table to graph the line $y=\frac{1}{5}x-2.$

What is the difference between the equations of a vertical and a horizontal line?

## Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

After reviewing this checklist, what will you do to become confident for all objectives?

how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?